439

Appendix ASummary of Vector and Tensor Notation

In general, we have used tensorial notation throughout the book. Tensors of rank 0(scalars) are denoted by means of italic type letters a; tensors of order 1 (vectors) bymeans of boldface italic letters a and tensors of rank two and higher orders by cap-ital boldface letters A. In some special circumstances, three-dimensional Cartesiancoordinates are used:

a.ai / vector;

A.Aij/ tensor of rank 2;

U.ıij/ unit tensor .ıij is Kronecker’s symbol/;

J.Jijk/ tensor of rank 3:

A.1 Symmetric and Antisymmetric Tensors

Denoting by superscript T the transpose, the symmetric and antisymmetric tensorsare respectively defined as

symmetric A D AT .Aij D Aji/; antisymmetric A D �AT .Aij D Aji/: (A.1)

The trace of a tensor is defined as the sum of its diagonal components, namely

trace of a tensor Tr A DX

i

Aii : (A.2)

A.2 Decomposition of a Tensor

It is customary to decompose second-order tensors into a scalar (invariant) part A, a

symmetric traceless part0

A , and an antisymmetric part Aa as follows

440 A Summary of Vector and Tensor Notation

A D 13.Tr A/UC 0

A CAa D 13Aıij C

0

AijCAaij : (A.3)

Note that this decomposition implies Tr0

A D 0 .Pi

0

Aii D 0/.The antisymmetric part of the tensor is often written in terms of an axial vector

aa whose components are defined as

aai D

Xj;k

"ijkAajk; (A.4)

where the permutation symbol "ijk has the values

"ijk D

8ˆ<ˆ:

C1 for even permutations of indices .i:e: 123; 231; 312/

�1 for odd permutations of indices .i:e: 321; 132; 213/

0 for repeated indices.

(A.5)

A.3 Scalar (or Dot) and Tensorial (Inner) Products

We have used for the more common products the following notation:Dot product between

two vectors a � b DPi

aibi .scalar/;

a vector and a tensor A � b DPj

aijbi .vector/;

a tensor and a vector b � A DPj

bjajk .vector/;

two tensors A � B DPk

aikbkj .tensor/:

(A.6)

Double scalar product between tensors

A W B DXi;k

aikbkj .scalar/: (A.7)

The trace of a tensor may also be written in terms of its double scalar product withthe unit matrix as TrA D A W U.

A.6 Differentiation 441

A.4 (Inner) Tensorial Product (also Named Dyadic Product)

between two vectors .ab/ij D aibj .tensor of rank 2/;

a vector and a tensor .aB/ijk D aiBjk .tensor of rank 3/;

a tensor and a vector .Ba/ijk D Bijak .tensor of rank 3/;

two tensors .AB/ijkl D AijBkl .tensor of rank 4/:

(A.8)

A.5 Cross Multiplication Between Two Vectorsand Between a Tensor and a Vector

.a � b/k D "ijkaibj .vector/;

.B � a/ik DXj;l

"jklBijbl .tensor/: (A.9)

A.6 Differentiation

The most usual differential operators acting on tensorial fields may be expressed interms of the so-called nabla operator, defined in Cartesian coordinates as

r D�@

@x1

;@

@x2

;@

@x3

�: (A.10)

Gradient (defined as dyadic product)

.ra/i D @a

@xi

.vector/; .ra/ij D @aj

@xi

.tensor of rank 2/;

.rA/jki D @Ajk

@xi

.tensor of rank 3/:

Divergence (defined as the scalar product)

r � a DX

i

@ai

@xi

.scalar/; .r � A/i DX

j

@Aji

@xj

.vector/: (A.11)

Rotational or curl (defined as the cross product)

.r �a/i DXj;k

"ijk@ak

@xj

.vector/; .r �A/ik DXj;l

"ijk@Aij

@xl

.tensor of rank 2/:

442 A Summary of Vector and Tensor Notation

The most usual second-order differential operator in tensorial analysis is theLaplacian, defined as

r � r DX

i

@2

@xi@xi

: (A.12)

A.7 Tensor Invariants

Some combinations of the elements of a tensor remain invariant under changes ofcoordinates. Such invariant combinations are

I1 D TrA D A W U DX

i

Aii;

I2 D TrA � A D A W A DXi;j

AijAji; (A.13)

I3 D TrA � A � A DXi;j;k

AijAjkAki:

Other invariant combinations may also be formed, but they are combinations of I1,I2 and I3; for instance, one often finds the invariants I , II and III defined as

I D I1; II D 12.I 2

1 �I2/; III D 16.I 3

1 �3I1I2C2I3/ D det A: (A.14)

The invariants I , II and III appear as coefficients in the “characteristic equation”det.�U� A/ D 0:It is also possible to form joint invariants of two tensors A and B as

I11 D Tr A �B; I21 D Tr A �A � B; I12 D Tr A �B �B; I22 D Tr A �A �B �B:(A.15)

Appendix BUseful Integrals in the Kinetic Theory of Gases

We present here some useful integrals appearing in several calculations based onthe kinetic theory of gases. Let F.C / be any scalar function of the peculiar velocityC such that the integrals appearing below converge, and let Cx and Cy be twocomponents of C . Then

ZF.C /C 2

x dC D 1

3

ZF.C /C 2dC ; (B.1)

ZF.C /C 4

x dC D 1

5

ZF.C /C 4dC ; (B.2)

ZF.C /C 2

xC2y dC D 1

15

ZF.C /C 4dC : (B.3)

The following definite integrals are also useful

Z 1

0

exp.�˛C 2/C rdC Dp�

2

1

2

3

2

5

2: : :

r � 12

˛�.rC1/=2 .r even/; (B.4)

Z 1

0

exp.�˛C 2/C rdC D 1

2Œ.r � 1/=2�Š˛�.rC1/=2 .r odd/: (B.5)

443

445

Appendix CSome Physical Constants

Boltzmann’s kB 1:38 � 10�23 J K�1 D 8:62 � 10�5 eV K�1

constantStefan-Boltzmann’s �0 5:67 � 10�8 W m�2 K�4

constantRadiation constant a D 4�0=c 7:56 � 10�16 J m�3 K�4

Atomic mass unit amu 1:66 � 10�27 kgElectron charge e 1:60 � 10�19 CElectron mass me 9:11 � 10�31 kgProton mass mp 1:673 � 10�27 kgPlanck’s constant h 6:63 � 10�34 J s D 4:14 � 10�15 eV s

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Author Index

A

Abramo, L.R.W., 431Ackerson, B.J., 187Alder, B.J., 285Alley, W.E., 285Alsemeyer, H., 269Alvarez, F.X., 234, 237, 238, 240Andreev, N.S., 295, 296Anile, A.M., 118, 156, 221, 240, 254, 260,

261, 269, 270, 334–337Antaki, P.J., 206–208Archimedes, 317Arrhenius, 206, 208Asheghi, M., 237Astarita, G., 366, 367

B

Baccarani, G., 335, 336Balescu, 113Bali, R., 429Banach, Z., 80, 148, 414Baranyai, A., 81, 184, 190Barrow, J.D., 426, 427, 430Barton, G., 338Basedow, 395–397Bataille, J., 347Bauer, H.-J., 262Bauschinger, 359Bedeaux, D., 3, 215Bekenstein, 432Belinskii, V.A., 425, 427, 429Benard, 35Beris, A.N., 32, 33, 399Berne, B.J., 169, 276Bidar, H., 150, 189, 419Binder, K., 297

Bird, R.B., 31, 359, 360, 362, 363, 365, 373,385

Bloch, 338Blotekjaer, K., 335Boillat, G., 85, 414Boltzmann, 8, 77, 79, 93–97, 99, 101,

103–107, 111, 113, 124, 129, 134,138, 144, 164, 170, 210, 234, 236,240, 241, 245, 249–251, 269, 301,327, 334, 360, 361, 363, 413, 414,417, 418, 430, 445

Boon, J.P., 276, 280, 284, 287Bose, 130, 156, 161, 162, 225, 243, 426Boukary, M.S., 31Brenner, 316Bressan, A., 413Brey, 80, 113, 133Brillouin, 279–282, 334Brown, 300, 360Burnett, 31

C

Cahn, J.W., 291, 293–397Callaway, 210Callen, H.B., 125Calvao, M.O., 426Camacho, J., 84, 110, 161, 162, 300, 329, 365Carnot, 8, 76Carrassi, M., 254, 258–260Carreau, P.J., 366Casas-Vazquez, J., 75, 77–79, 152, 397Casimir, H.B.G., 21, 135, 245Castillo, V.M., 214Cattaneo, C., 42, 44, 46, 47, 50–53, 58, 59,

62, 63, 200–202, 204–209, 211,218, 233, 234, 239–241, 244, 245,248–251, 259, 280, 303–306, 347,350, 358

469

470 Author Index

Cauchy, 6, 349Cavalli-Sforza, L.L., 306Chapman, S., 15, 93, 99, 100Chen, G., 74, 234, 240, 245, 246, 248, 250,

251Chester, M., 201Christiansen, R.F., 374Christoffel, 415Chu, 254Chung, C.-H., 285, 287Ciancio, V., 340Cimmelli, V.A., 80Clark, N.A., 187, 288Clause, 113Clausius, 8, 23, 24, 28–31, 45, 47, 60, 72–74Cloot, A., 254, 259–261Cole, 343Coleman, B.D., 3, 23, 26, 28, 30, 216, 218, 347Compte, A., 307, 308Conforto, F., 340Corbet, A.B., 148Coriolis, 21, 31, 137Couette, 81, 88, 113, 151, 152, 363, 365Cowling, T.G., 93, 99Crank, J., 302Criado-Sancho, M., 47, 81, 86, 152, 387, 388,

391, 392, 396, 402Criminale, W.O., 368Crisanti, A., 75, 81Crochet, M.J., 368Curie, 18

D

Daivis, P.J., 184Danielewicz, P., 418Daniels, E., 175, 287Danilov, D.A., 320, 321D’Anna, G., 81Dauby, P.C., 376Davalos-Orozco, L.A., 341Davies, P.C.W., 435, 459Day, W., 30Debye, 161, 201, 218, 243, 340, 342Dedeurwaerdere, T., 217De Felice, L.J., 331De Gennes, P.G., 31De Groot, S.R., 3, 8, 16, 309, 333, 340Del Castillo, L.F., 341, 343, 402, 403De Leener, M., 169Dempsey, J., 338Denbigh, R.G., 3

Denicol, G.S., 419de Sitter, 433Dirac, 130, 131, 331Di Stefano, V., 338Doi, M., 359Domınguez, R., 83, 146, 153, 156Doppler, 84Dreyer, W., 217, 414Drucker, D.C., 357–359Dufour, 37, 311Dufty, 105Duhamel, 350Duhem, P., 3, 24, 28, 29, 31, 60Dunn, J.E., 367, 371Durning, C.J., 303–305Dynes, R.C., 218

E

Eckart, C., 3, 407, 408, 410, 433Eddington, 156, 221Edelen, D.G.D., 31Edwards, S.F., 359Einstein, 123, 124, 129, 130, 135, 156, 161,

162, 164, 225, 243, 330, 424–426,431

Enskog, 15, 99, 100Ericksen, J.L., 31, 347, 367–371Eringen, C., 349Ernst, 105, 133Eu, B.C., 48, 74, 99, 110–113, 143, 148Euclid, 5, 25, 26, 64Euler, 4, 7, 308, 338, 356Evans, D.J., 139, 183–188, 190Eyring, 113

F

Fai, G., 418Falcon, N., 435Faraudo, J., 83, 156Fedotov, S., 306, 307Fermi, 79, 130, 331, 338, 340Fick, 19, 137, 138, 291, 293, 301–304, 306,

311, 315, 402Fisher, D.J., 320, 321Flory, 385, 387, 388, 392, 402Fokker, 299Fort, J., 79, 306–308Fosdick, R.L., 367, 371

Author Index 471

Fourier, 19, 22, 23, 30, 42, 43, 50, 51, 53, 58,116, 179, 201, 203–210, 220–222,233–236, 239–241, 244, 245, 250,255, 264–266, 269, 276–278, 282,307, 315, 335, 336, 342, 343, 350,410

Fratzl, P., 297Friedmann, 424, 425, 429, 434Friedrichs, 85Frisch, H.L., 301Frohlich, H., 162Furth, 297

G

Galenko, P., 296, 297, 319–321Galileo, 5, 6, 12Garcıa-Colın, L.S., 48, 80, 114, 288Gariel, J., 431Garrido, P.L., 234Garzo, V., 104Gauss, 5, 13, 126, 131, 137, 138, 184, 288,

395Germain, 357Ghosh, K., 138, 149Giambo, S., 340Giannozzi, P., 117Giardina, R., 234Gibbons, G.W., 433Gibbs, J.W., 3, 15, 22, 30, 32, 48, 49, 51–54,

56, 62–64, 74, 82, 85, 93, 102, 103,115, 123, 125–127, 143, 146, 149,151, 186, 212, 292–294, 308–310,313, 315, 316, 318, 328, 331, 332,341, 349, 352, 355, 369, 375, 384,385, 393, 395, 407, 418, 431

Giesekus, H., 348, 364, 365, 369, 375–376Ginzburg, 34, 293Goldstein, H., 158Goldstein, S., 297Gorban, A.N., 103Grabert, H., 169Grad, H., 31, 93, 99–103, 110, 114, 149, 153,

214, 263, 267, 269–271, 414, 415,417

Grandy, W.T., 143Gray, 106Greco, A., 225Green, A.E., 106, 107, 127, 174, 184, 202,

331, 348, 349, 368Greenspan, M., 254, 260, 261Griffin, 417

Grmela, M., 4, 32, 33, 35, 74, 113, 214, 215Gurney, W.S.C., 148Guyer, R.A., 208–211, 222, 223, 234Gyarmati, I., 3, 48

H

Haase, R., 3Hafskjold, B., 184Halperin, B.I., 338Hamilton, 3, 32, 33, 83, 113, 145, 157, 158,

170, 172, 293, 383Hanley, H.J.M., 184–187Hansch, W., 334, 335Harris, 100Hatano, 81Hawking, S.W., 432, 433Hayward, G., 426Helmholtz, 8, 24, 125, 186, 212, 292, 294,

385Herlach, D., 224, 233, 321Hermite, 100, 149, 171, 174, 270Herrera, L., 435Hess, S., 118, 184Hilliard, J.E., 291, 293–397Hiscock, W.A., 407, 411–413, 419, 426, 430Holway, L.H., 270Hooke, 157, 358–360, 363Hoover, 79, 187Hoover, W.G., 31Hoover, W.H., 184Hoover, W.M., 214Hopfenberg, H.B., 301, 304Hubble, 425, 427–429, 433Hu, B.L., 426Huggins, 385, 387–389, 392, 402Hugoniot, 265–268

I

Ichiyanagi, M., 148Israel, W., 134, 407, 408, 414, 416, 417

J

Jackson, H., 218Jacobi, 33–35Jahnig, E., 165

472 Author Index

Jaumann, 27, 64Jaynes, E.T., 138, 143Jeffrey, A., 215, 234, 353, 354, 358, 376Johri, V.B., 431Jones, 113, 183, 186Joseph, D.D., 201Joshi, A.A., 204, 205, 234, 240, 241, 243,

244Jou, D., 48, 75, 77–79, 81, 83, 84, 110, 130,

131, 133, 146, 148, 150, 152, 153,161, 162, 201, 221, 225, 234, 237,238, 240, 253, 262, 270, 284, 286,291, 300, 328, 329, 331, 365, 383,385, 402, 419, 426, 430

Joule, 8, 328Junk, M., 338Juttner, 414

K

Karlin, I.V., 103Kausch, 384, 395Kawasaki, 383Keizer, J., 80, 81, 123, 131, 165Kelvin, 347, 353, 354Kelvin, Lord, 8, 76Kestin, J., 65, 347Keyes, 343Kirchhoff, 255, 256, 258, 349Kivelson, 343Kjelstrup, S., 3, 5, 184, 215Kluitenberg, G., 347Knesser, H.O., 262Knudsen, 233, 234, 239, 240, 250Kohlrausch, 329Koide, T., 419Kozlowski, 419Krall, A.N., 328Kramers, 299, 364Kranys, M., 262, 414Kremer, F., 162Kreuzer, 3Kronecker, 127Kroner, E., 359Krumhansl, J.A., 208–211, 222, 223,

234Kubo, 127, 174, 184, 331Kurz, W., 320, 321

L

Lagrange, 6, 13, 29, 31, 60–62, 64, 143–146,149–154, 159–161, 164, 189, 348

Lambermont, J., 15, 82, 354, 359Lame, 353Landau, L.D., 4, 34, 125, 129, 131, 224, 265,

266, 280, 293, 407, 408Langevin, 173Laplace, 12, 179, 180, 202, 254, 258, 266, 277,

278, 282, 284, 342, 343, 442Larecki, W., 156Larson, B.C., 237, 240, 399Larson, P.M., 157, 161Lavenda, B., 3Lax, P.D., 85, 268Lebedev, V., 296, 297Lebon, G., 3, 15, 31, 48, 52, 56, 61, 65,

82, 213, 214, 225, 254, 259–261,268, 292, 294, 306, 309, 328, 352,374–376, 402

Lebowitz, J.L., 157Le Denmat, G., 431Legendre, 52, 138, 355, 356Lennard, 113, 183, 186Leppard, W.R., 374Lepri, S., 235, 238, 239Levermore, C.D., 156Levermore, D., 219, 221Levine, R.D., 143Lhuillier, D., 311, 314, 316, 317Li, B., 237, 430Lifshitz, E.M., 125, 131, 265, 266, 293, 407Lima, J.A.S., 426, 427, 431Lindblom, L., 407, 411–413Liouville, 169–171, 174Liu, I.-S., 29, 31, 60, 61, 130, 214, 407, 414Liu, W., 237Llebot, J.E., 332Lodge, A.S., 366, 374Loose, W., 184, 383Lorentz, 12, 21, 137, 156, 221, 279–281Lotka, 306Luikov, A.V., 201, 202Lumley, J., 31Luzzi, R., 81, 143, 144, 147

M

Maartens, R., 430MacDonald, 399–402

Lyapunov, M., 43

Author Index 473

Mach, 224, 266, 268–270Machlup, S., 48, 131Madureira, J., 180Majorana, A., 269, 270Majumdar, A., 204, 205, 234, 240, 241,

243–245Markov, 298Marshak, 249,Martınez, J., 435Marucci, G., 366, 367Mashelkar, 399Masoliver, J., 297Maugin, G., 65, 311, 347, 357, 383Mavrantzas, 32, 33, 399Maxwell, J.C., 8, 22, 42, 58, 59, 72, 73, 80,

97, 127, 129, 152, 170, 186, 200,201, 240, 259, 260, 280, 283, 288,303–306, 347, 353, 354, 358, 362,368, 371, 385, 388, 398, 435

Mayer, 8Mazur, P., 3, 8, 16, 309, 333, 340McLennan, J.A., 31, 105McQuarrie, D.A., 169Meixner, J., 3, 17, 80, 347Mendez, V., 306, 307Metzner, 373Meyer, E., 254, 260, 261Mihalas, 156Miller, B.N., 22, 157, 161Mongiovı, M.S., 225, 253Monte Carlo, 251, 327, 335–337Morgan, 213Mori, 173, 179, 284, 341Morriss, G.P., 183, 188Morro, A., 254, 258–260Moyal, 299, 354Muller, I., 31, 48, 60, 80, 86, 130, 225, 258,

268, 288, 370, 399–402Murdoch, 31Murphy, G.L., 426Muscato, O., 334, 335, 338Muschik, W., 80, 347, 383

N

Nagano, 116Naqvi, K.R., 240, 241Narayanamurti, V., 218Navier, 23, 260, 261, 264–266, 269, 283, 284,

286, 371Neogi, P., 303, 304Nettleton, R.E., 48, 59, 79, 83, 148, 186

Neumann, 350Newman, D., 216, 218Newton, 19, 27, 30, 58, 65, 84, 211, 255,

312–314, 347, 353, 354, 358, 361,366, 367, 369, 370, 375, 423–425

Nguyen, 384, 394Nicolis, G., 43Nigmatulin, R.Y., 312, 317Nisbet, R.M., 148Noll, W., 3, 18, 23, 26, 347Nose, 184Nozieres, 398Nyquist, 331

O

Ohm, 19, 328Oldroyd, 368, 376Oliveira, H.P., 426, 427Olson, T.S., 419Onsager, L., 3, 20–22, 30, 34, 42, 48, 57, 123,

131, 135–137, 309–311, 315, 327,331–333, 336, 342, 350

Onuki, A., 383, 384, 398Ortiz de Zarate, J.M., 162Ostwald, 366, 372, 373Ottinger, H.C., 4, 32, 34, 35, 113, 383, 399Owen, 30Ozisik, M.N., 201, 243

P

Pankratov, O., 338, 339Pavon, D., 131, 133, 270, 407, 410, 426, 427,

429–432, 434, 435Pecora, R., 276Peebles, P.J.E., 433Peltier, 333Pennisi, S., 334, 335, 337Perez-Garcıa, C., 284, 286, 287, 361Perzyna, P., 354Peshkov, 201Piola, 349Placzek, 280Planck, 3, 23, 98, 124, 130, 144, 299, 424, 426,

432, 445Pluchino, S., 118, 240, 254, 260, 261Poiseuille, 190, 191, 222Poisson, 33–35, 113, 114, 170, 335, 338Pomraning, 246

474 Author Index

Ponter, A.R.S., 354Poynting, 347, 353, 354Pradhan, A., 429Prager, W., 354Prakash, 399Preziosi, L., 201Prigogine, I., 3, 15, 43, 214, 431

Q

Qiu, 224Quemada, 398

R

Rangel-Nafaile, 383, 385, 387–389, 391Rankine, 265, 267, 268Rayleigh, 279, 280, 282Reichl, L.E., 105Reik, H.G., 3Reiner, M., 366, 367, 370, 371Resibois, P., 169Reynolds, O., 6, 13, 22Ricci, 424Rice, 106Richter, P.H., 165Ritort, F., 75, 81Rivlin, R.S., 31, 347, 366–371Robertson, B., 118, 143, 147, 424Rodrıguez, R.F., 341Romano, V., 337, 338, 430, 432Rosenau, P., 299Ross, J., 301Rouse, P.E., 348, 359–361, 363, 365, 375, 385,

388–393Ruggeri, T., 60, 86, 214, 267–270, 414Russo, G., 337Ryskin, 31

S

Sackur, 98, 103Salamon, P., 48, 201Salim, J.M., 426, 427Salmonson, 426, 430Santos, A., 80, 104, 113Sato, H., 131Schlesinger, M.F., 329

Schlogl, F., 143Schmitz, R., 165Schultz, B.E., 424, 426, 433Schweizer, M.A., 301, 414Searles, D.J., 139Seebeck, 333Seeger, A., 359Seitaridou, E., 137, 138Senger, J.V., 162Sessler, G., 254, 260, 261Sharts, 219Sieniutycz, S., 48, 201Silhavy, M., 30Smoluchowski, 300Sobolev, S.L., 224Solderhom, 241Soret, 37, 311, 315Spohn, H., 157Stannett, V., 301Stefan, 134, 245, 445Stewart, J.M., 134, 413, 414, 416, 417Stocker, H., 417Stokes, 19, 23, 28, 30, 58, 211, 260, 261,

264–266, 269, 279, 283–286, 328,329, 359, 361, 371, 423

Struchtrup, H., 118, 213–215, 217, 414Strumia, 85Sudharan, P., 431Sussman, R.A., 431

T

Tabor, M., 303–305Taniuti, T., 215Tanner, 372Taylor, G.I., 212, 291, 297, 299Tetrode, 98, 103Thomas, N.L., 303–305Thomson, N.L., 333, 347, 353, 354, 411Thomson, William, 76Tien, 224Todd, B.D., 190Tokatly, I.V., 338, 339Torrilhon, M., 214, 215Toupin, R., 5, 26Tremblay, A.M.S., 162, 163, 165, 331Tribus, M., 143Triginer, J., 431, 432Trivelpiece, A.W., 328Troughton, H., 260Truesdell, C., 3, 5, 8, 18, 21, 23, 26Turok, N., 426Tzou, D.Y., 201, 223, 224, 234, 243

Author Index 475

U

Udey, N., 416Uehling, 418Uhlenbeck, 418

V

Valenti, A., 211, 220, 268, 269van den Hoogen, R.J., 430van Weert, 413Vasconcellos, A.R., 144Velasco, R.M., 114, 288Verhas, J., 340Vernotte, P., 201Vidal, F., 331Vlad, M.O., 301Vlasov, 338Voigt, 347, 353, 354, 358Volterra, 306von Mises, 357Vostner, 118

W

Waldenstrom, A., 240Waldmann, L., 118, 215Wald, R.M., 424

Walker, 424Wards, 201Watts, 329Weinberg, S., 424, 426Weissenberg, 366Weiss, W., 213, 214, 269, 270, 288, 414Wilks, 201Williams, 329Wilmanski, K., 85, 370Wilson, 293Windle, A.H., 303–305Wolf, 387, 389Wong, 133Woods, L.C., 3, 8, 29, 100, 260Wordeman, M.R., 335, 336

Y

Yang, R., 251Yip, S., 276, 280, 284, 285, 287

Z

Zakari, M., 221, 426, 430Zhang, Z.M., 234Zienkiewics, 213Zimdahl, W., 430, 432Zimm, B.H., 348, 359–361, 365, 375, 385,

388–393Zubarev, D.N., 143, 147Zwanzig, 284

Subject Index

A

Absolute temperature, 86, 329, 341, 360classical, 15, 23, 54non-equilibrium, 54, 71, 73, 76, 78, 140,

146, 153, 161, 167Absorption coefficients, 255, 258, 260, 261Acceleration, waves, 26, 42, 215–218,

262–264, 304, 317, 410, 419Admissible process, 24Affinity, 16, 36, 393–394Angular velocity, 21, 26, 27, 31, 37, 68,

137, 230Anomalous diffusion, 321–322Astrophysics, 407, 423, 435Availability, 125

B

Balance laws, 4–12, 24, 28, 29, 41, 61, 266,292, 319

of charge, 11, 340of energy, 4, 11, 49, 54, 60, 68, 247, 272,

313, 340, 369of entropy, 54, 313, 369of mass, 4, 9, 11, 54, 272, 295, 313of momentum, 4, 11, 272, 313

Ballistic transport, 233, 234, 245, 252, 337Big-bang singularity, 426Biological membranes, 331Black holes, 426, 432Boltzmann equation, 79, 93–95, 97, 99, 103,

104, 119, 170, 210, 226, 234, 236,241, 249, 250, 327, 334, 414, 417,418

entropy, 98, 288Boltzmann-Planck formula, 124

Bose-Einstein statistics, 130Boundary conditions, 18, 157, 206, 213, 214,

243, 249, 252Brillouin peaks, 279–282, 289Bulk viscous pressure, 8, 41, 99, 256, 280,

384, 408, 417, 423, 424, 429, 434viscosity, 134

Burnett equations, 31, 290

C

Cahn-Hilliard’s model, 294–297Calortropy, 74, 111Carnot’s cycle, 39Carreau’s model, 366Casimir limit, 245, 252Cattaneo equation, heat transport, 201, 202,

208–221, 227, 248Cattaneo’s law, 47, 51, 52, 58, 59, 62, 204, 205,

233, 239, 241, 244, 245, 248–251,259, 305

Chapman-Enskog expansion, 100Characteristic speeds, 85, 299, 304, 411–413Charged systems, 10–12Chemical potential, 85, 130, 137, 295, 305,

313, 328, 344, 384–388classical, 15, 36, 146, 394, 401non-equilibrium, 71, 146, 195, 345, 384,

386, 395, 402–404, 409Chemical reactions, 8–10, 15–17, 19, 35–36,

38, 205, 206, 208, 262, 291, 292,306, 308, 351, 383, 393–397, 404

Christoffel symbols, 415Classical irreversible thermodynamics, 3,

14–23, 82, 93, 118, 123, 192, 215,275, 309, 310, 327, 341, 347

Clausius-Duhem inequality, 24, 28, 29, 31, 60Clausius-Planck inequality, 23

477

478 Subject Index

Coldness, 80Collision time, 111, 114, 119, 222, 259, 370,

427, 436Computer simulations, 71, 79, 83, 183–195,

234, 235, 275, 276, 282, 285, 336Concavity of the entropy, 15, 64, 85, 309,

375Conduction current, 11Configurational distribution function, 363Configuration tensor, 383Constitutive equations, 17–24, 27–32, 37, 38,

56, 57, 60, 66, 68, 99, 190, 194, 209,223, 290, 314, 315, 317, 331–333,335, 336, 350, 351, 366, 369, 370,376, 379, 399, 425

Contact temperature, 80Continued-fraction expansions, 114–119,

179–181, 252Convected derivatives, 27Convected Maxwell’s model, 378, 379,

385, 398Coriolis force, 21, 31, 137Corotational time derivative, 56Correlated random walk, 291, 297–299Cosmic scale factor, 424Cosmological horizons, 423–438Cosmology, 35, 407, 423, 425, 430Couette flow. See Shear flowCritical, Mach number, 270

Crystallization front, 324Curie’s principle, 18Current density, 10, 277, 329, 330, 344

D

Dark energy, 423, 429Dark matter, 423, 429Debye’s model of solids, 89, 166, 225Degeneracy requirements, 33, 34Degenerate gases, 130Degradation of polymers, 393Degree of coupling, 38Dendritic patterns, 320–322Density correlation function, 275–277Detailed balance, 36Dielectric relaxation, 340–343Diffusion, 9, 11, 16, 19, 36, 43, 81, 87, 137,

138, 141, 142, 149, 202–204, 209,221, 224, 226, 234, 240, 245–251,

260, 291–325, 335, 341, 345, 354,384, 392, 397–403, 405

case II, 302–305Diffusion coefficient, 81, 142, 293, 299, 303,

315, 324, 345, 392, 398, 400, 401,403

Dissipation, 17, 29, 32, 35, 59, 62, 81, 140,147, 152, 167, 173, 184, 221, 314,316, 351, 352, 354–356, 359, 407,423, 426

Dominant energy condition, 433, 435, 438Drift-difffusion model, 335Drude’s relation, 331Dufour’s law, 37, 311Dumbbell solutions, 376–378Dynamical temperature, 69, 80

E

Eckart’s reference frame, 408Eddington’s factor, 156, 221Effective relaxation times, 114, 117–118, 217Effective thermal conductivity, 220, 236, 238,

239, 251, 252Efficiency of energy conversion, 38Einstein’s formula for diffusion coefficient, 81

equation (general relativity), 345, 424formula for fluctuations, 130, 135, 330

Electrical systems, 327–346Energy-momentum tensor, 408, 423Entropy

classical, 43, 44, 46, 47, 102, 123, 194,330, 437

extended, 44, 52, 141flux, classical, 179flux, extended, 46, 48, 76, 138, 161, 221,

260, 275, 290flux fourvector, 414, 430flux, kinetic, 57, 75, 209information, 56, 88, 126, 144, 147–149,

414, 420microscopic definition, 98production, 13–17, 20, 22, 32, 37, 39, 43,

47–57, 60, 64, 67, 68production, generalized, 55–57

Equations of state, non-equilibrium, 71–89,143, 185–188, 384, 399, 403, 418,419

Equilibrium constant, 394, 395, 404, 405Equipartition, breaking of, 165Equipresence principle, 24, 25, 63Euclidean transformations, 25–27

point, 387

Subject Index 479

Evolution equations for the fluxes, 41, 71, 93,101, 109, 270, 285, 294, 305, 308,310, 314, 317, 318, 336, 409, 410,417, 419

Extensional flow, 379Eyring’s formula, 113

F

Fermi-Dirac statistics, 130, 331Fick’s law, 19, 137, 138, 142, 291, 293, 301,

311First law of thermodynamics, 5, 45Flory-Huggins’ formula, 385Fluctuation-dissipation theorem, 81, 152, 173Fluctuations, equilibrium, 75, 80, 123–128,

130under an electric field, 81of the fluxes, 82, 126, 127under a heat flux, 89, 166under a shear flow, 394

Fluxes, 7, 41, 71, 93, 123, 144, 169, 211, 235,253, 275, 292, 327, 355, 399, 408

Flux fluctuation theorems, 137–139, 165Flux limiters, 219–221, 437Fokker-Planck equation, 299Forces (thermodynamic), 3–70, 77, 104, 135,

146, 174, 214, 292, 310, 314, 317,327, 332, 333, 348, 355, 359–363,399

Fourier’s law, 19, 42, 43, 50, 53, 201, 204, 205,209, 210, 220, 221, 235, 252, 315,350

Fourth-order terms in the entropy, 67, 344Frame indifference principle, 25–28, 31, 56,

68, 367Friedmann-Robertson-Walker metric, 424

G

Galilean invariance, 5Gaussian distribution, 126Generalised Newtonian fluid, hydrodynamics,

222Generating functional, 32GENERIC formulation, 311Gibbs’ equation, classical, 146

for electrical conductors, 341for fluids, 15generalized, 54, 63, 195, 227

for heat conductors, 52for mixtures, 15for viscoelastic bodies, 194

Gibbs’ free energy, 32, 292, 385, 403, 404Giesekus’ four-parameter model, 369, 375Ginzburg-Landau equation, 34Grad’s method, 100, 103, 114Green-Kubo formulae, 127Guyer-Krumhansl’s equation, 210, 222, 223,

226, 228

H

Hamiltonian thermodynamics, 3–40, 383Harmonic chain, 157–162, 166Heat conductivity. See Thermal conductivityHeat conductors, 59–63, 86–88, 230, 233–241,

293, 302flux, 16, 42, 52, 56, 57, 69, 86, 157, 237,

293pulses, 202, 217waves, 47, 70, 225

Helmholtz free energy, 24, 38, 212, 385Hermite polynomials, 100, 110, 120, 149, 270Hierarchy of evolution equations, 181, 235Higher-order fluxes, 66, 74, 114, 118, 179,

259, 288–289, 331, 340H -theorem, 96–99, 119, 120, 148, 182,

300–301Hubble’s factor, 425, 429, 437Hyperbolic heat transport, 87, 199–231

I

Ideal gas, 36, 65, 78, 79, 81–84, 87–89, 96,100, 128–130, 141, 147–157, 165,166, 225, 235, 262–264, 266, 270,273, 283, 362, 378, 399, 404, 410

Ignition, 205–208, 306Inertia of heat, 42, 410Inflationary expansion, 428Information theory, 143–167Integrability conditions, 51, 74, 212, 230, 341Interfaces, 93, 224, 228, 229, 233, 294,

319–321, 323Internal degrees of freedom, 65, 110, 223, 262,

280, 281Internal energy, 5–7, 9–12, 14, 23, 24, 28, 42,

48, 49, 53, 60, 69, 71, 77, 79, 80, 84,89, 95, 96, 124, 140, 144, 151, 154,

480 Subject Index

156, 157, 185, 186, 194, 211–213,229, 230, 242, 245, 247, 249, 254,263, 308, 313, 327, 328, 340, 344,345, 348, 349, 351, 352, 360, 363,378, 408, 420

Internal variables theory, 69, 311, 316, 318Irreversibility, 14

J

Jacobi’s identity, 33–35Jaumann’s time derivative, 27, 64, 68, 370, 379Jeffrey’s model, 353, 376Jump condition, 5

K

Kelvin-Voigt’s model, 347, 379Kinetic constants, 36, 395Kinetic theory

of gases, 15, 31, 56, 87, 88, 93–121, 130,165, 259, 271, 331, 361, 362, 383,432, 443

of polymers, 359–365Knudsen number, 233, 234, 239, 240, 250Kohlrausch-Williams-Watts law, 329Kramers-Moyal’s expansion, 299

L

Lagrange multipliers, 29, 31, 60–62, 64,143–146, 149–154, 159–161, 164,167, 189, 226

Lame’s coefficient, 353Landau–Lifschitz

hydrodynamic noise, 131reference frame, 265

Landau–Placzek’s ratio, 280Langevin’s equation, 140, 173Laser, 81, 113, 202, 205, 208, 220, 223, 224,

228, 289, 301, 351Legendre’s transform, 52, 138, 195, 355, 356Light scattering, 42, 163, 173, 253, 275,

278–280, 282, 288Linear response, 169–182Liouville’s equation, 170Local equilibrium hypothesis, 14–15, 22, 47,

81, 103, 323

Longitudinal velocity correlation function,282, 285–287

Lorentz force, 11, 12, 21, 137Lyapunov time, 43

M

Mach number, 224, 266, 268–270, 273Magnetic field, 21, 137Mass action law, 35, 36, 39, 395Mass fraction, 9, 11, 14, 16, 20, 292, 295, 308,

312, 313, 318, 396Master equation, 182Material indifference. See Frame indifference

principleMaximum-entropy formalism, 111, 124,

138, 147, 153, 163, 189, 214, 221,335

Maxwell-Boltzmann’s distribution, 97, 129,152, 362

Maxwell-Cattaneo equations, 58, 59, 68–70,73, 127, 170, 195, 200, 201,227, 280, 303–305, 435. See alsoCattaneo equation, heat transport

Maxwell’s viscoelastic model, 283, 347, 353,354

Mean free path, 18, 23, 88, 103, 140, 157,165, 184, 221, 233, 235–238,243–245, 249, 251, 258, 264, 269,270, 272, 273, 289, 331, 337, 345,426, 436

Memory functions, 68, 118, 169, 170, 173,174, 177, 180, 181, 276, 284, 285,287

Memory principle, 25Metals, 202, 224, 227, 262, 272Metric tensor, 408, 420, 424Microelectronic devices, 327, 337Microfluidic system, 137, 149Micropolar fluids, 37Minimum entropy production, 39, 214, 226Mixtures, 8–10, 15, 84, 133, 140, 292–297,

308–312, 314, 321, 384, 388, 395,396, 410, 426, 436, 437

Modified moment method, 111, 112Molecular dynamics, 71, 81, 113, 183–185,

192, 282, 283, 405Moments of the distribution, 100, 110, 235,

334, 335Monte Carlo simulations, 251, 327, 336, 337

Subject Index 481

N

Nanosystems, 199, 208, 233, 275Nanowires, 223, 237, 252Neutrinos, 133, 407, 431, 435–437Neutron scattering, 23, 42, 163, 173, 175, 253,

275, 280, 282Newtonian fluids, 84, 222, 312–314, 354, 361,

366, 369, 375Newton-Stokes’ law of viscosity, 19, 30, 58,

211Non-Fickian diffusion, 291, 301–305, 316Non-ideal gases, 66, 93, 106–110, 139Non-linear transport, 110–114, 183Non-Newtonian fluids, 84, 347, 348, 354,

365–376Normal modes, 360, 363, 365Normal stress coefficients, 366, 372, 374, 375,

379Nuclear collisions, 71, 79, 421Nuclear equation of state, 417, 419Nyquist formula, 331

O

Objective quantities, 27Objective time derivatives, 27, 31, 264, 368,

370Objectivity, 25, 26, 64Ohm’s law, 19, 328Oldroyd’s model, 368, 376Onsager-Casimir’s relations, 20–22, 34,

135–137, 315Ostwald power law, 366, 373

P

Pair-correlation function, 106, 139, 187Partition function, 145, 149, 150, 154, 159,

160, 166, 319Phase diagrams under flow, 383–388,

390Phenomenological coefficients, 18–20, 31, 38,

39, 50, 53, 56, 68, 82, 210, 211,214, 236, 238, 260, 309–311, 315,350, 352

Phonon hydrodynamics, 199, 218, 222–223,252

Phonon radiative transfer, 204, 234, 241–245

Photons, 133, 140, 156, 167, 219, 221, 243,301, 410, 420, 426, 436

Plasmas, 81, 113, 219, 220, 328, 334–340,407, 417, 419

Plasticity, 348, 354–359Poisson brackets, 4, 32–35, 40, 113, 114,

170Polymers, 42, 65, 66, 85, 103, 107, 110, 141,

193, 264, 291, 301–305, 347, 348,354, 359–365, 371, 383–406

Poynting-Thomson’s model, 347, 353, 354,379, 380

Pressure, non-equilibrium, 54, 71, 73, 75, 76,82–85, 185–188, 392, 421

Pressure tensor, 6, 8–10, 16, 21, 24, 27, 37, 38,54, 63, 65, 82–84, 96, 100, 105–107,111, 129, 139, 141, 148, 150, 156,174–176, 178, 187, 188, 190, 191,225, 260, 262, 273, 280, 290, 313,314, 333, 347, 348, 355, 361, 365,368, 384, 385, 398, 399, 405, 408,414, 418

Projection operators, 133, 169–174, 178, 179,284

Propagation speed of signals, 199, 201, 407,423, 425

R

Radiation hydrodynamics, 246Radiative gas, 133–134, 167Random force, 172, 173Random walk, 291, 297–299, 346Rankine-Hugoniot equation, 265, 267, 268Rational extended thermodynamics, 41,

59–64Rational thermodynamics, 3, 23–32, 35, 38,

59, 347, 367, 371Rayleigh’s peak, 279, 280, 282Reaction-diffusion systems, 291, 306–308,

324Reciprocal relations. See Onsager-Casimir’s

relationsReiner-Rivlin fluid, 366, 367, 371Relativistic formulation of extended

thermodynamics, 407–409, 414, 419Relativistic ideal gas, 154–157Relaxation-time approximation, 104–106, 120,

139, 226, 236, 241, 328Relaxation times of the fluxes, 41, 58, 106,

114, 163, 174, 253, 275Representation theorems of tensors, 29

482 Subject Index

Rheological models, 369, 373, 376Ricci’s tensor, 424Rivlin-Ericksen’s fluid, 31, 367–371Rotating cylinder, 230Rotational viscosity, 37Rouse-Zimm’s model, 361, 385, 388–383,

392

S

Sackur-Tetrode’s formula, 98, 103Second law of thermodynamics, 13, 28, 45,

349, 371, 434Second-order fluid, 367, 368Second sound, 67, 199–202, 217, 225, 230,

231, 413Semiconductors, 81, 331, 334–340Shear flow, 84, 141, 150–153, 166, 187,

188, 190, 191, 193, 367, 377–379,383–406

Shear modulus, 284Shear rate, 81, 84, 88, 107, 113, 119, 183–191,

193, 194, 347, 363, 366, 368, 369,372–375, 377, 378, 383, 385, 386,388, 390–393, 396, 397, 403, 404,421

Shear thinning, 113, 189Shock waves, 42, 253–273Solidification front, 291, 324Soret law, 37, 311, 315Sorption experiments, 301, 302Sound propagation, 253–262, 280, 413Specific heat, 15, 80, 85, 126, 141, 162, 176,

225, 319, 330Spectrum of density fluctuations, 277, 279Spinodal decomposition, 293–297Spinodal line, 386–391Stability conditions, 67, 186, 187, 383, 398,

413Steady-state compliance, 385, 388, 406Stefan-Boltzmann’s law, 134, 245Stochastic noise, 131Stochastic processes, 291, 297–301Stoichiometric coefficients, 9, 36, 393Stokes’ law, 19, 58, 285, 328, 329Stretched exponential decay, 329Subtracted heat flux, 128Superfluids, 42, 201, 224–225, 253

Suspensions, 36, 42, 71, 185, 186, 291–325,398

Symmetric hyperbolicity, 85

T

Taylor’s dispersion, 291Telegrapher’s equation, 86, 200–202, 224,

228, 239, 294, 297–301, 306, 308,323

Temperature. See Absolute temperatureThermal conductivity, 19, 58, 89, 112, 116,

119, 139, 140, 166, 174, 202, 221,223, 225, 227, 228, 236, 241, 287,333, 407, 418, 419

effective, 220, 236, 238, 239, 251, 252,336

Thermodiffusion, 21, 22Thermoelasticity, 348–351Thermometer, 71, 78, 81, 84, 192Third law, 162Thirteen-moment approximation. See Grad’s

methodThomson’s relation, 333Time-reversal invariance, 21, 59Transverse velocity correlations function,

282–285, 289, 290Transverse waves, 273Turbulence, 31, 225Two-layer model, 323

U

Ultrarelativistic gas, 410, 413Ultrasound propagation in gases, 42, 240,

253Uncompensated heat, 73

V

Variational principles, 213Velocity gradient, 7, 28, 37, 38, 56, 68, 104,

183, 184, 191, 230, 291, 303, 347,361–363, 367, 368, 377–379

Viscoelastic fluids, 194Viscoelasticity, 351–354, 358, 360

Subject Index 483

Viscometric functions, 348, 366, 367,371–374, 379

Viscositybulk, 19, 30, 58, 134, 140, 237, 255, 262,

267, 407, 423, 425–427, 430–433,436, 437

rotational, 37shear, 19, 20, 30, 58, 112, 119, 127, 139,

140, 189, 284, 289, 353, 375, 377,401, 418, 426

Viscous fluids, 63–64, 88–89, 283, 321,413

pressure tensor, 63Vlasov regime, 338Volterra functional derivative, 33

W

Wavesin fluids, 253–273in plasmas, 338

Weissenberg’s effect, 366Williams-Watts’s law, 329

Z

Zeroth law, 75–79, 190–192